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We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the weights, we establish a permuton limit theorem describing the global behavior of Luce-distributed permutations and derive an explicit density of the limiting permuton. We further compute limiting pattern densities and analyze the differences between exact Luce permutations and their permuton approximations. We also study the local convergence of these permutations, proving a quenched Benjamini--Schramm limit and a central limit theorem for consecutive pattern occurrences. Finally, we prove a central limit theorem for the number of inversions.

Originally introduced by Kolmann and Shelah as a surrogate for saturated models, limit models have been established as natural and useful objects when studying abstract elementary classes. Shelah began the study of when (multiple notions of) limit models exist for first order theories. In this paper we look at their structure. In superstable theories it is known that all limit models are isomorphic, but in the strictly stable case the number of non-isomorphic limit models was not well understood. Here we characterise the full spectrum of limit models in the first order stable setting, by a short and simple argument using only the familiar machinery of stable first order theories: $\textbf{Theorem.}$ Let $T$ be a complete $λ$-stable theory where $λ\geq |\operatorname{L}(T)| + \aleph_0$. Let $δ_1, δ_2 < λ^+$ be limit ordinals where $\operatorname{cf}(δ_1)< \operatorname{cf}(δ_2)$. Let $N_l$ be a $(λ, δ_l)$-limit model for $l = 1, 2$. Then $N_1$ and $N_2$ are isomorphic if and only if $\operatorname{cf}(δ_1) \geq κ_r(T)$. Moreover, if $κ_r(T) = \aleph_α$, there are exactly $|α| + 1$ limit models up to isomorphism. In the context of first order stable theories, this reduces the p

A novel decoupling limit of the membrane is proposed, leading to the $(1+2)$-dimensional classically integrable model originally introduced by Manakov, Zakharov, and Ward. This limit is the large-wrapping regime of a membrane propagating toy background of the form $\mathbb{R}_t \times T^2 \times G$ subject to scaling limit, where $G$ is a Lie group and the geometry is supported by a four-form flux. Such toy backgrounds can arise from consistent eleven-dimensional supergravity solutions, exemplified by the uplift of the pure NSNS AdS$_3 \times$ S$^3 \times$ T$^4$ background. The scaling limit can be interpreted as similtaneous small tension and non- or hyper-relativistic limit.

We prove the uniqueness of high cofinality limit models in stable abstract elementary classes (AECs) with amalgamation, assuming the existence of a rather weak independence relation. $\textbf{Theorem.}$ Suppose $\mathbf{K}$ is a $λ$-stable AEC, where $\operatorname{LS}(\mathbf{K}) \leq λ$, $κ< λ^+$ is regular, and $\mathbf{K}_λ$ satisfies the amalgamation property. Let $\mathbf{K}'$ is the class of all $(λ, δ)$-limit models where $\operatorname{cf}(δ) \geq κ$ (or any AC where $\mathbf{K}' \subseteq \mathbf{K}_λ$ contains all such $(λ, δ)$-limit models when $\operatorname{cf}(δ) \geq κ$). Suppose also that there is an independence relation on $\mathbf{K}'$ satisfying weak uniqueness, weak existence, universal continuity* in $\mathbf{K}_λ$, $(\geq κ)$-local character, and $(λ, θ)$-weak non-forking amalgamation in some regular $θ\in [κ, λ^+)$. Let $δ_1, δ_2 < λ^+$ be limit with $\operatorname{cf}(δ_l) \geq κ$ for $l = 1, 2$. Then for all $M, N_1, N_2 \in \mathbf{K}_λ$, if $N_l$ is $(λ, δ_l)$-limit over $M$ for $l = 1, 2$, then $N_1 \underset{M}{\cong} N_2$. Moreover, if $K_λ$ also satisfies the joint embedding property, then for all $N_1, N_2 \in \mathbf{K}_λ$, if $N_l$ is $(λ,

We study language generation in the limit under bounded memory. In this task, a learner observes examples from an unknown target language one at a time and must eventually output only new valid examples. Prior work assumes access to the entire history, a strong assumption since realistic algorithms retain limited past information. Classical work in learning theory shows memory constraints dramatically alter learnability; we extend this to language generation. First, we study memoryless generators. Under a mild enumeration restriction, every countable collection of infinite languages remains generable without memory. Without this restriction, we exactly characterize when memoryless generation is possible. For finite collections, we characterize the optimal minimax density achievable by memoryless generators -- the best density guaranteed against any collection of a given size. This combinatorial bound relies on Sperner's theorem and symmetric chain decompositions. We further show that a sliding window of the last $W$ examples does not improve this worst-case density, whereas allowing it to store $b$ adaptively chosen past examples improves the achievable density for every $b \geq 1$

Let $M_i$ be a sequence of non-collapsed $n$-manifolds with two-sidedly bounded Ricci curvature. We show that the Gromov-Haudorff limit space, $Y$, of the associated sequence of orthonormal frame bundles, $FM_i$, equipped with an almost canonical metric, shares similar properties as a Ricci limit space of non-collapsing sequence i.e., the singular set has codimension $\ge 4$ whose complement contains an open and dense $C^{1,α}$-Riemannian manifold.

We study the set of normalized multi-lengths for representations of closed surface groups and free groups into $(\mathrm{PSL}_2\mathbf{R})^d$ whose projections to $\mathrm{PSL}_2\mathbf{R}$ are all convex cocompact. These multi-lengths define a convex cone in $\mathbf{R}^d_{\geq 0}$, called the limit cone. When $d=3$, we show the coexistence of different regimes: for some representations the limit cone has only a finite number of sides, which we can force to grow like the genus (or free rank); for other representations, extremal rays are dense in the boundary of the limit cone. We also give examples where the limit cone varies discontinuously with the representation.

Forecasting accuracy is bounded by the information available about the future. This paper makes that statement precise using information-theoretic tools. Under logarithmic loss, the expected performance of any probabilistic forecast decomposes into two parts: an irreducible component and an approximation component. The irreducible term is the conditional entropy of the future given the available information, while the approximation term is the divergence between the true conditional distribution and the forecasting method. The gap between this conditional-entropy limit and an unconditional baseline is exactly the mutual information between the future observation and the declared information set. This leads to a definition of forecastability as the maximum achievable reduction in expected log loss. Evaluated across horizons, forecastability forms a profile that describes how predictive information varies with lead time. This profile reflects the dependence structure of the process and need not be monotone: predictive information may be concentrated at particular lags, including seasonal horizons, even when intermediate horizons contain little useful signal. From this profile, the pa

We prove that the long-run behavior of Hawkes processes is fully determined by the average number and the dispersion of child events. For subcritical processes we provide FLLNs and FCLTs under minimal conditions on the kernel of the process with the precise form of the limit theorems depending strongly on the dispersion of child events. For a critical Hawkes process with weakly dispersed child events, functional central limit theorems do not hold. Instead, we prove that the rescaled intensity processes and rescaled Hawkes processes behave like CIR-processes without mean-reversion, respectively integrated CIR-processes. We provide the rate of convergence by establishing an upper bound on the Wasserstein distance between the distributions of rescaled Hawkes process and the corresponding limit process. By contrast, critical Hawkes process with heavily dispersed child events share many properties of subcritical ones. In particular, functional limit theorems hold. However, unlike subcritical processes critical ones with heavily dispersed child events display long-range dependencies.

In this work, we experimentally investigate the frequency limit of Hall effect sensor designs based on a 2 dimensional electron gas (2DEG) gallium arsenide/aluminum gallium arsenide (GaAs/AlGaAs) heterostructure. The frequency limit is measured and compared for four GaAs/AlGaAs Hall effect sensor designs where the Ohmic contact length (contact geometry) is varied across the four devices. By varying the geometry, the trade-off in sensitivity and frequency limit is explored and the underlying causes of the frequency limit from the resistance and capacitance perspective is investigated. Current spinning, the traditional method to remove offset noise, imposes a practical frequency limit on Hall effect sensors. The frequency limit of the Hall effect sensor, without current spinning, is significantly higher. Wide-frequency Hall effect sensors can measure currents in power electronics that operate at higher frequencies is one such application.

We show that the winding of low-lying closed geodesics on the modular surface has a Gaussian limiting distribution when normalized by any standard notion of length, in contrast to the Cauchy distribution arising when allowing arbitrarily deep excursions into the cusp. In addition, we prove a Berry-Esseen bound and a local limit theorem.

This paper analyzes weak solutions of the quantum hydrodynamics (QHD) system with a collisional term posed on the one-dimensional torus. The main goal of our analysis is to rigorously prove the time-relaxation limit towards solutions to the quantum drift-diffusion (QDD) equation. ewline The existence of global in time, finite energy weak solutions can be proved by straightforwardly exploiting the polar factorization and wave function lifting tools previously developed by the authors. However, the sole energy bounds are not sufficient to show compactness and then pass to the limit. ewline For this reason, we consider a class of more regular weak solutions (termed GCP solutions), determined by the finiteness of a functional involving the chemical potential associated with the system. For solutions in this class and bounded away from vacuum, we prove the time-relaxation limit and provide an explicit convergence rate. ewline Our analysis exploits compactness tools and does not require the existence (and smoothness) of solutions to the limiting equations or the well-preparedness of the initial data. ewline As a by-product of our analysis, we also establish the existence of global in

We study the scaling limit of statistical mechanics models with non-convex Hamiltonians that are gradient perturbations of Gaussian measures. Characterising features of our gradient models are the imposed boundary tilt and the surface tension (free energy) as a function of tilt. In the regime of low temperatures and bounded tilt, we prove the scaling limit with respect to infinite volume Gibbs states for macroscopic functions on the continuum, and we show that the limit is a continuum Gaussian Free Field with covariance (diffusion) matrix given as the Hessian of surface tension. Our proof of this longstanding conjecture for non-convex energy complements recent studies in [Hil16, ABKM], as well as the proof for strictly convex Hamiltonians in [AW22].

In this paper we provide a method to generate a continuum of limit cycles using a single precomputed exponentially stable limit cycle designed within the Hybrid Zero Dynamics framework. Guarantees for existence and stability of these limit cycles are provided. We derive analytical constraints that ensure boundedness of the state under arbitrary switching among a finite set of limit cycles extracted from the continuum. These limit cycles are used for changing the speeds of an underactuated planar bipedal model while satisfying all modeling constraints such as saturation torque and coefficient of friction in a provably correct manner. A strongly connected directed graph of allowable limit cycle switches is built to obtain valid limit cycle transitions for speed changes within 0.42-0.81 m/s.

We present a new self-consistent perturbative expansion for realistic isolated differentially rotating systems -- disc galaxies. At leading order it is formally equivalent to Ehlers' Newton-Cartan limit, which we reinterpret in terms of quasilocal energy and angular momentum. The self-consistent coupling of these quasilocal terms leads to first-order differences from the conventional Newtonian limit. A modified Poisson equation is obtained, along with modifications to the equations of motion for the effective fluid elements. By fitting to astrophysical data, we show that the phenomenology of collisionless dark matter for disc galaxies can be reproduced. Potential important consequences for gravitational physics on galactic and cosmological scales are briefly discussed.

Given an image generated by the convolution of point sources with a band-limited function, the deconvolution problem is to reconstruct the source number, positions, and amplitudes. This problem arises from many important applications in imaging and signal processing. It is well-known that it is impossible to resolve the sources when they are close enough in practice. Rayleigh investigated this problem and formulated a resolution limit, the so-called Rayleigh limit, for the case of two sources with identical amplitudes. On the other hand, many numerical experiments demonstrate that a stable recovery of the sources is possible even if the sources are separated below the Rayleigh limit. In this paper, a mathematical theory for the deconvolution problem in one dimension is developed. The theory addresses the issue when one can recover the source number exactly from noisy data. The key is a new concept "computational resolution limit" which is defined to be the minimum separation distance between the sources such that exact recovery of the source number is possible. This new resolution limit is determined by the signal-to-noise ratio and the sparsity of sources, in addition to the cutof

We study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population convergence of the corresponding stochastic process. For this purpose, we use an individual-based model whose state space explicitly takes into account the duration of infectious periods. This allows to deal with the natural correlation of the epidemic states of individuals whose household and workplace share a common infected. In a general setting where a non-exponential distribution of infectious periods may be considered, convergence to the unique deterministic solution of a measurevalued equation is obtained. In the particular case of exponentially distributed infectious periods, we show that it is possible to further reduce the obtained deterministic limit, leading to a closed, finite dimensional dynamical system capturing the epidemic dynamics. This model reduction subsequently is studied from a numerical point of view. We illustrate that the dynamical system derived from the large population approximation is a pertinent model reduction when

The quantum speed limit indicates the maximal evolution speed of the quantum system. In this work, we determine speed limits on the informational measures, namely the von Neumann entropy, maximal information, and coherence of quantum systems evolving under dynamical processes. These speed limits ascertain the fundamental limitations on the evolution time required by the quantum systems for the changes in their informational measures. Erasing of quantum information to reset the memory for future use is crucial for quantum computing devices. We use the speed limit on the maximal information to obtain the minimum time required to erase the information of quantum systems via some quantum processes of interest.

We present limits on $ν_μ(\barν_μ)\toν_τ(\barν_τ)$ and $ν_μ(\barν_μ)\toν_e(\barν_e)$ oscillations based on a study of inclusive $νN$ interactions performed using the CCFR massive coarse grained detector in the FNAL Tevatron Quadrupole Triplet neutrino beam. The sensitivity to oscillations is from the difference in the longitudinal energy deposition pattern of $ν_μN$ versus $ν_τN$ or $ν_e N$ charged current interactions. The $ν_μ$ energies ranged from 30 to 500 GeV with a mean of 140 GeV. The minimum and maximum $ν_μ$ flight lengths are 0.9 km and 1.4 km respectively. For $ν_μ\toν_τ$ oscillations, the lowest 90% confidence upper limit in $\sin^22α$ of $2.7\times 10^{-3}$ is obtained at $Δm^2\sim50$~eV$^2$. This result is the most stringent limit to date for $25<Δm^2<90$ eV$^2$. For $ν_μ\toν_e$ oscillations, the lowest 90% confidence upper limit in $\sin^22α$ of $1.9\times 10^{-3}$ is obtained at $Δm^2\sim350$~eV$^2$. This result is the most stringent limit to date for $250<Δm^2<450$ eV$^2$, and also excludes at 90% confidence much of the high $Δm^2$ region favored by the recent LSND observation.